WAVELET ANALYSIS OF CONSERVATIVE CASCADESSIDNEY RESNICK, GENNADY SAMORODNITSKY, ANNA GILBERT, AND WALTER WILLINGERAbstract. A conservative cascade is an iterative process that fragments a given set into smallerand smaller pieces according to a rule which preserves the total mass of the initial set at eachstage of the construction almost surely and not just in expectation. Motivated by the importanceof conservative cascades in analyzing multifractal behavior of measured Internet tra�c traces,we consider wavelet based statistical techniques for inference about the cascade generator , therandom mechanism determining the re-distribution of the set's mass at each iteration. We providetwo estimators of the structure function, one asymptotically biased and one not, prove consistencyand asymptotic normality in a range of values of the argument of the structure function less thana critical value. Simulation experiments illustrate the asymptotic properties of these estimatorsfor values of the argument both below and above the critical value. Beyond the critical value,the estimators are shown to not be asymptotically consistent.1. IntroductionA multiplicative cascade is an iterative process that fragments a given set into smaller andsmaller pieces according to some geometric rule and, at the same time, distributes the total massof the given set according to another rule. The limiting object generated by such a proceduregenerally gives rise to a singular measure or multifractal { a mathematical construct that is able tocapture the highly irregular and intermittent behavior associated with many naturally occurringphenomena, e.g., fully developed turbulence (see [10, 12, 4, 15] and references therein); spatialrainfall [6]; the movements of stock prices [14]; and Internet tra�c dynamics [19, 3].The generator of a cascade determines the re-distribution of the set's total mass at everyiteration; it can be deterministic or random. Cascade processes with the property that thegenerator preserves the total mass of the initial set at each stage of the construction almost surelyand not just in expectation are called conservative cascades and are the main focus of this paper.Originally introduced by Mandelbrot [13] (also in the turbulence context), conservative cascadeshave recently been considered in [3] for use in describing the observed multifractal behavior ofmeasured Internet tra�c traces. In particular, Feldmann et al. [3] build on empirical evidencethat measured Internet tra�c is consistent with multifractal behavior by illustrating that \:::data networks appear to act as conservative cascades!" They demonstrate that� multiplicative and measure-preserving structure becomes most apparent when analyzingmeasured Internet traces at a particular layer within the well-de�ned protocol hierarchyof today's Internet Protocol (IP)-based networks, namely the Transport Control Protocol(TCP) layer and at the level of individual TCP connections,The visits of Sidney Resnick to AT&T Labs{Research were supported by AT&T Labs{Research and a NationalScience Foundation Grant from the Cooperative Research Program in the Mathematical Sciences. S. Resnick andG Samorodnitsky were also partially supported by NSF grant DMS-97-04982 and NSA grant MDA904-98-1-0041at Cornell University. 1

2 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER� this structure is recovered at the aggregate level (i.e., when considering the superpositionprocess consisting of all IP packets generated by all active TCP connections) and causesaggregate Internet tra�c to exhibit multifractal behavior.Well short of providing a physical explanation to the all-important networking question of \Whydo packets within individual TCP connections conform to a conservative cascade?" the work in [3]is empirical in nature and relies on a number of heuristics for inferring multifractal behavior fromtraces of measured Internet tra�c. However, to provide a more solid statistical basis for empiricalstudies of multifractal phenomena, progress in the area of statistical inference for multiplicativelygenerated multifractals is crucial.In this paper we contribute to the e�ort of providing rigorous techniques for multifractalanalysis by investigating wavelet-based estimators for conservative cascades (i.e., for the classof multifractal processes generated by conservative cascades) and studying their large sampleproperties. In essence, the inference problem for conservative cascades consists of deducing froma single realization of the cascade process the distribution of the cascade generator that waspresumably used to generate the sample or signal at hand. Intuitively, the generator's distributioncan be inferred from the degree of variability and intermittency exhibited locally in time by thesignal under consideration. It can be expressed mathematically in terms of the local H�olderexponents which in turn characterize the singularity behavior of a signal locally in time. Moreover,since the local H�older exponent at a point in time t0 describes the local scaling behavior of thesignal as we look at smaller and smaller neighborhoods around t0, a wavelet-based analysis thatfully exploits the time- and scale-localization ability of wavelets proves convenient and is tailor-made for our purpose. On the one hand, we exploit here the fact that the singularity behavior ofa process can (under certain assumptions) be fully recovered by studying the singularity behaviorin the wavelet domain; i.e., by investigating the (possibly) time-dependent scaling properties ofthe wavelet coe�cients associated with the underlying process in the �ne-time scale limit. On theother hand, using Haar wavelets, the discrete wavelet transform of a conservative cascade can beexplicitly expressed in terms of the cascade's generator (see for example [5]) and hence providesa promising setting for relating the local scaling behavior of the sample to the distribution ofthe underlying conservative cascade generator. In particular, we relate the distribution of thegenerator to an invariant of the cascade, namely the structure function or modi�ed cumulantgenerating function (also known as Mandelbrot-Kahane-Peyri�ere (MKP) function [7]) and studythe statistical properties (i.e., asymptotic consistency, asymptotic normality, con�dence intervals)of two wavelet-based estimators of this function.Although the results in this paper have been largely motivated by our empirical investigationsinto the multifractal nature of measured Internet tra�c [3, 5], we have clearly bene�ted fromthe recent random cascade work of Ossiander and Waymire [17]. Compared to the conservativecascades considered in this paper, random cascades are multiplicative processes with generatorsthat preserve the total mass of the initial set only in expectation and not almost surely. Thisapparently minor di�erence ensures independence within and across the di�erent stages of a ran-dom cascade construction but gives rise to subtle dependencies inherent in conservative cascades.Ossiander and Waymire [17] study the large sample asymptotics of estimators that are de�ned inthe time-domain rather than in the wavelet-domain and allow for a rigorous statistical analysisof the scaling behavior exhibited by random cascades (for related work, see also [20]). Whilethe large-sample properties of the time domain-based estimators considered in [17] and of the

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 3wavelet-based estimators studied in this paper are very similar, their potential advantages, dis-advantages and pitfalls when implementing and using them in practice require further studies.However, in combination, these di�erent estimators provide a set of statistically rigorous tech-niques for multifractal analysis of highly irregular and intermittent data that are assumed to begenerated by certain types of multiplicative processes or cascades.The rest of the paper is organized as follows. Sections 2{4 contain the basic facts about conser-vative cascades, their wavelet transforms, and some related quantities that are needed later in thepaper. Section 5 discusses the critical constants and Section 6 is concerned with certain martin-gales and leads into Section 7 where subcritical asymptotics (that is, asymptotics for values of theargument below the critical value) and strong consistency of our two wavelet-based estimatorsis established. Asymptotic normality of the estimators is explained and illustrated with somesimulated data in Section 8. We conclude in Section 9 with some supercritical asymptotics whenthe value of the arguement exceeds the critical value. The values of the estimators at large valuesof the argument of the structure function are uninformative and misleading, thus providing somepractical guidance for properly interpreting the plots associated with the estimation procedure.2. The Conservative Cascade.We now summarize the basic facts about the conservative cascade.Consider the binary tree. Nodes of the tree at depth l will be indicated by (j1; : : : ; jl) 2 f0; 1gl:Alternatively we consider successive subdivisions of the unit interval [0; 1]. After subdividing ltimes we have equal subintervals of length 2�l indicated byI(j1; : : : ; jl) = h lXk=1 jk2k ; lXk=1 jk2k + 12l�; (j1; : : : ; jl) 2 f0; 1gl:(2.1)An in�nite path through the tree is denoted byj = (j1; j2; : : : ) 2 f0; 1g1and the �rst l entries of j are denoted byjjl = (j1; : : : ; jl):We will sometimes write when convenientjjl; jl+1 = (j1; : : : ; jl; jl+1):The conservative cascade is a random measure on the Borel subsets of [0; 1] which may beconstructed in the following manner. Suppose we are given a random variable W , called thecascade generator , which has range [0; 1] and which is symmetric about 1/2 so that W d= 1�W:The symmetry implies that E(W ) = 1=2: We assume the random variable is not almost surelyequal to 1/2. There is a family of identically distributed random variablesfW (jjl); j 2 f0; 1g1; l � 1geach of which is identically distributed as W . These random variables satisfy the conservativeproperty W (jjl; 1) = 1�W (jjl; 0):(2.2)Random variables associated with di�erent depths of the tree are independent and random vari-ables of the same depth which have di�erent antecedents in the tree are likewise independent.

4 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERDependence of random variables having the same depth is expressed by (2.2). The conservativecascade is the random measure �1 de�ned by�1(I(jjl)) = lYi=1W (jji):(2.3)Note the conservative property entails that�1(I(jjl; 0)) + �1(I(jjl; 1)) = �1(I(jjl));(2.4)so that the weight of two o�spring equals the weight of the parent. This impliesXjjl �1�I(jjl)� = 1:(2.5) 3. Wavelet Coefficients.We compute the wavelet transformd�l;n = Z 10 �l;n(x)�1(dx); n = 0; : : : ; 2l � 1; l � 1;(3.1)using the Haar wavelets �l;n(x) := (2l=2; if 2n2l+1 � x < 2n+12l+1 ;�2l=2; if 2n+12l+1 � x < 2n+22l+1 :(3.2)We have by examining where the Haar wavelet is constant thatd�l;n = 2l=2��1�[ 2n2l+1 ; 2n+ 12l+1 )�� �1�[2n+ 12l+1 ; 2n+ 22l+1 )�� :Now suppose thatPlk=1 jk=2k = n=2l: Then we have from the de�nition (2.3)d�l;n =2l=2h lYi=1W (jji)W (jjl; 0)� lYi=1W (jji)W (jjl; 1)i=2l=2 lYi=1W (jji)hW (jjl; 0)�W (jjl; 1)iand using the conservative property (2.2), this isd�l;n =2l=2 lYi=1W (jji)h2Wjjl; 0)� 1i;(3.3)for n = 0; 1; : : : ; 2l � 1: Sometimes where convenient, we will also writed�l;n = d(�l; jjl) = 2l=2 lYi=1W (jji)h2Wjjl; 0)� 1i:(3.4)

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 54. Notation Glossary.Before continuing the analysis, we collect some notation in one place for easy reference. Weseek to estimate the distribution of the cascade generator W and this will be accomplished if weestimate c(q) := 2E(W q); q > 0;(4.1)or equivalently we could estimate the structure function�(q) = 1 + log2E(W q) = log2 c(q):(4.2)The structure function will be estimated using estimators constructed from the processZ(q; l) =Xjjl lYi=1W (jji)qj2W (jjl; 0)� 1jq(4.3)and note from (3.3) that Z(q; l) = 12ql=2 2l�1Xn=0 jd�l;njq:(4.4)Our analysis rests on the process M(q; l), which we will show to be a martingale and which isde�ned as M(q; l) = 1c(q)l Xjjl lYi=1W (jji)q; q > 0; l � 1;(4.5)and note the normalization makes E(M(q; l)) = 1:There are further constant functions needed:b(q) =Ej2W � 1jq;(4.6) a(q) =c(2q)c2(q) = E(W 2q)2�E(W q)�2 ;(4.7) ar(q) =c(rq)cr(q) = 21�rE(W rq)�E(W q)�r :(4.8)Note that a(q) = a2(q): Finally we need three variances�21(q) := 1c2Var(W q + (1�W )q);(4.9) �22(q) := 1b2Var(j2W � 1jq):(4.10) �23(q) := 1b2Var�W q1c j2W2 � 1jq + (1�W1)qc j2W3 � 1jq � j2W1 � 1jq�;(4.11)where Wi; i = 1; 2; 3 are iid with the distribution of the cascade generator.It is convenient to de�ne W = e�Y so that the Laplace transform of Y is�(q) := Ee�qY = E(W q)(4.12)

6 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERand ar(q) = 21�r�(rq)�r(q) :5. Critical ConstantsWe now de�ne the quantity q� := supfq > 0 : a(q) < 1g(5.1)so that for q < q� we have a(q) < 1: It will turn out that when q < q�, the sequence fM(q; l); l � 1gis an L2-bounded and uniformly integrable martingale and this is the easiest case to analyze. Itis always the case that q� � 1, which follows from the fact thata(1) = E(W 2)2(E(W ))2 = E(W 2)2(12)2 = 2E(W 2)so that a(1) = 2�Var(W ) + 14� = 2E(W � 12)2 + 12 � 2 � j1� 12 j2 + 12 = 1:The Mandelbrot-Kahane-Peyri�ere (MKP) Condition: Let W be the cascade generatorand de�ne Xq = W qEW q ; q > 0;so that EXq = 1: The MKP Condition is satis�ed for q ifE(Xq log2Xq) < 1(5.2)i� qE(W q)E(W q logW )� logE(W q) < log 2(5.3)i� q(log �)0(q)� log � < log 2 :(5.4)De�ne �� := fq : E(Xq log2Xq) < 1g:Then �� is an interval and we de�ne the second critical constantq� := sup��:(5.5)Why is q� considered a critical quantity? It turns out that the martingale fM(q; l); l � 1gconverges as l!1 to M(q;1) whereM(q;1) = ( 0; if q � q�;something non-degenerate; if q < q�:It turns out that the associated martingale is uninformative asymptotically when q > q�.The two critical constants are related numerically by the inequalitymax(1; q�=2) � q� � q�:(5.6)

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 7To see why the inequality q� � q� in (5.6) is true, it su�ces to show that if q > 0 satis�es a(q) < 1,then the MKP condition is satis�ed for this q. However, if a(q) < 1, thenlog 2 > log �(2q)� 2 log �(q) = log �(2q)� log �(q)� log �(q)=Z 2qq (log �)0(s)ds� log �(q)and since log � is convex, (log �)0 is increasing and the forgoing is bounded below by�(log �)0(q)Z 2qq ds� log �(q)=q(log �)0(q)� log �(q):We conclude log 2 > q(log �)0(q)� log �(q)which is equivalent to the MKP condition holding by (5.4).On the other hand, suppose that q� <1. Since a(q�) = 1, we have in the same way as abovelog 2 = log �(2q�)� 2 log �(q�) = 2�log �(2q�)� log �(q�)�� log �(2q�)=2Z 2q�q� (log �)0(s)ds� log �(2q�)�2(log �)0(2q�)Z 2q�q� ds� log �(2q�)=2q(log �)0(2q�)� log �(2q�):Therefore, the MKP condition does not hold for 2q�, and so q� � 2q�.Example 1. Suppose W is uniformly distributed on [0; 1]. In this case E(W q) = 1=(1 + q) andso a(q) = 12 �1 + q22q + 1�and q� = 1 +p2 � 2:4:Likewise, q� satis�es the equation log(1 + q)� q1 + q = log 2and so q� � 3:311:Example 2. Suppose more generally that W has the beta distribution with mean 1/2 (i.e., theshape parameters � and � are equal). ThenE(W q) = �(2�)�(�+ q)�(�)�(2�+ q)(5.7)and q� satis�es4��p��(�+ 2q�)�(2�+ q�)2 � �(�+ 1=2)�(2�+ 2q�)�(�+ q�)2 = 0:

8 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERExample 3. Suppose W has the two point distribution concentrating mass 1=2 at �p for some0 � p < 1=2. Then E(W q) = 12 (pq + (1� p)q)(5.8)and a(q) = 1� 2pq(1� p)q(pq + (1� p)q)2 :Note in this case that a(q) " 1 as q " 1 so q� = q� =1.Example 4. If W does not have a two point distribution but nevertheless has an atom of sizep1 < 1=2 at 1 (and hence by symmetry there is an atom of the same size at 0) we have q� < 1(and, hence, also q� < 1). To see this we express the condition (5.3), when q > 1, in theequivalent form E(W q log2W q)E(W q) log2�2E(W q)� > 1 :(5.9)Note that if W does not have a two point distribution, then for q > 1, we have P [W q < W ] > 0and E(W q) < E(W ) = 1=2, so log2(2E(wq)) < 0, which explains the sign reversal in (5.9)compared with (5.3).By the dominated convergence theorem the numerator in (5.9) converges to 0 as q !1, whilethe denominator converges to P [W = 1] log2�2P [W = 1]� 6= 0. Hence, (5.9) fails for large q.Based on the experience provided in Examples 3 and 4, it natural to wonder how common itcan be that q� =1. This is discussed in the next proposition.Proposition 5.1. Unless W has a two point distribution, it must be the case that q� <1.Proof. Because of Example 3, we may assume that W does not have atoms at 0 and 1. Letp 2 [0; 1=2) be the leftmost point of the support of the distribution of W . Then 1 � p is therightmost point of the support of the distribution of W . For 0 < � < 1, we have�(�) := P [W � �(1� p)] > 0:Since the distribution of W is not a two point distribution, lim�!1 �(�) < 12 : Thus, we can �ndand �x a value of 0 < � < 1 such that�(�) < 1=2; 0 < � < 1:For this value of �, it is convenient to set�(�) := � = �(1� p):Apply Jensen's inequality with the convex function g(x) = x log x; x > 0; to getE�W qc(q) log2�W qc(q)� � 1[W��]� =�(�)E�W qc(q) log2 W qc(q) ���W � ����(�)E�W qc(q) ���W � �� log2�E�W qc(q) ���W � ���=E� W q2E(W q)1[W��]� � log2E� W q2E(W q)���W � ��:(5.10)

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 9Also we have limq!1E�W qc(q)1[W��]� = 12 :(5.11)To verify (5.11), note that12 = E�W qc(q)� = E�W q)c(q) 1[W��]�+E�W qc(q)1[W<�]�;and 0 � limq!1 E(W q1[W<�])E(W q1[W��]) � limq!1 1P [W � �]E�W� �q1[W=�<1] = 0;by dominated convergence. Thus from (5.10) and (5.11), we concludelim infq!1 E�W qc(q) log2 W qc(q) � 1[W��]� � 12 � log2� 12�(�)� =: h > 0;(5.12)because �(�) < 1=2:We also claim limq!1E����W qc(q) log W qc(q) ���1[W<�]� = 0:(5.13)To verify (5.13), note the expectation is the same asE����W qc(q) log W qc(q) ���1[Wqc(q)< �qc(q) ]�:Provided that limq!1 �q=c(q) = 0;(5.14)we get for any � < e�1, by the monotonicity of jx log xj in (0; e�1), that the expectation is boundedby j� log2 �j for q so large that �q=c(q) < �. So it remains to check (5.14), or equivalently to checklimq!1E�W q�q � = 0:However, by Fatou's lemmalim infq!1 E�W q�q � � E�lim infq!1 �W� �q1[W��]� =1;since P [W > �] > 0 (otherwise, the de�nition of p would be contradicted).Our conclusion from (5.12) and (5.13) is that for all large q,E�W qc(q) log2�W qc(q)�� = E�W qc(q) log2�W qc(q)��1[W��] + E�W qc(q) log2�W qc(q)��1[W<�] > 0:Thus E W q2EW q �log2 W qE(W q) � log2 2� = 12EXq log2Xq � 12 > 0and so the MKP condition (5.3) fails for all large q. Therefore, q� <1.

10 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER5.1. Properties of the function ar(q). We need the following properties of the function ar(q).Proposition 5.2. (i) For any �xed r > 1, the function ar(q) (and therefore a(q)) is strictlyincreasing in q > 0.(ii) For any �xed q > 0, the function log ar(q) is strictly convex in the region r > 1.(iii) If q satis�es the MKP condition, thenddr log ar(q)���r=1 < 0;and there exists r0 2 (1; 2) such thatar0(q) < a1(q) = 1:(5.15)(iv) If the MKP condition fails for q and the inequalities in (5.2), (5.3) or (5.4) are reversed tobecome strictly greater than, we haveddr log ar(q)���r=1 > 0;(5.16)and there exists 0 < r1 < 1 and ar1(q) < a1(q) = 1.Proof. (i) Recall the de�nition of � from (4.12). For �xed r > 1, if we di�erentiate with respectto q, we get ��(rq)�r(q)�0 = �r(q)r�0(rq)� �(rq)r�r�1(q)�0(q)�2r(q) :This is positive i� �(q)�0(rq) > �(rq)�0(q)or �0(rq)�(rq) > �0(q)�(q) :Since r > 1, it su�ces to show �0=� is strictly increasing which is true if its derivative is strictlypositive. The derivative is �(q)�00(q)� (�0(q))2�2(q)and this is strictly positive i� �(q)�00(q) > (�0(q))2;(5.17)that is i� E(e�qY )E(Y 2e�qY ) > �E(Y e�q=2Y � e�q=2Y )�2which follows from the Cauchy{Schwartz inequality.(ii) Fix q > 0 and check thatd2dr2 (log ar(q)) = q2�2(rq) ��00(rq)�(rq)� (�0(rq))2�which is positive by (5.17).(iii) For �xed q > 0, ddr log ar(q) = q(log �)0(qr)� log 2� log �(q);

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 11so that ddr log ar(q)���r=1 = ddrar(q)ar(q) �����r=1 = q(log �)0(q)� log �(q)� log 2 < 0:Since a1(q) = 1, we have ddrar(q)���r=1 < 0; a1(q) = 1:Hence there exists r0 2 (1; 2) such thatar0(q) < a1(q) = 1:(iv) If ddr log ar(q)���r=1 = ddrar(q)ar(q) �����r=1 = q(log �)0(q)� log �(q)� log 2 > 0;then since log a1(q) = 0, there exists r1 < 1 such thatlog ar1(q) < 0 or ar1(q) < 1:6. The Associated Martingale.In this section we study the properties of the process fM(q; l); l � 1g de�ned in (4.5) for each�xed q > 0. We de�ne the increasing family of �-�eldsFl := �fW (jjl); jjl 2 f0; 1glggenerated by the weights up to and including depth l.Proposition 6.1. For each q > 0, the familyf(M(q; l);Fl); l � 1gis a non-negative martingale with constant mean 1 such that M(q; l) converges almost surely toa limiting random variable M(q;1):M(q; l) a.s.! M(q;1); E(M(q;1) � 1:If the MKP condition fails for q, thenP [M(q;1) = 0] = 1;and if q satis�es the MKP condition, then E(M(q;1)) = 1 so thatP [M(q;1) > 0] = 1:Proof. The martingale property is easily established:E(M(q; l+ 1)jFl) = Xjjl;jl+1E�Qli=1W q(jji)cl W q(jjl; jl+1)c jFl�=Xjjl lYi=1W q(jji)cl Xjl+1 E�W q(jjl; jl+1)=c�=M(q; l)2 �E(W q)=c =M(q; l):

12 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERBy the martingale convergence theorem (eg, [18], [16]) a non-negative martingale always convergesalmost surely. The last statements follow by the methods of Kahane and Peyri�ere ([8]). See alsoPropositions 6.2 and 6.3.Example 5. Recall the example of the two point distribution of Example 3 of Section 4. In thiscase we have M(q; l) = 1 for all q > 0 and l � 1. For verifying this, the key observation is thatW q + (1�W )q = pq + (1� p)q:(6.1)Recall (5.8) and then observe for l > 1M(q; l) =Xjjl lYi=1�W q(jji)c � = Xjjl�1 l�1Yi=1�W q(jji)c �Xjl W q(jjl � 1; jl)c= Xjjl�1 l�1Yi=1�W q(jji)c �hW q(jjl � 1; 0) +W q(jjl � 1; 1)i=cand since W (jjl� 1; 1) = 1�W (jjl� 1; 0) we apply (6.1) to get= Xjjl�1 l�1Yi=1�W q(jji)c �hpq + (1� p)qi=c= Xjjl�1 l�1Yi=1�W q(jji)c � =M(q; l� 1):One can easily see that M(q; 1) = 1 and the assertion is shown.De�ne M(q; 0) = 1 and let the martingale di�erences bed(q; l) :=M(q; l)�M(q; l� 1); l � 1:For l > 1 we have from the de�nition of M(q; l) thatd(q; l) = Xjjl�1 l�1Yi=1�W q(jji)c �hW q(jjl � 1; 0) + (1�W (jjl� 1; 0)qc � 1i= Xjjl�1 l�1Yi=1�W q(jji)c �h�(jjl)i:(6.2)We now easily see E(d(q; l)jFl�1) = 0. For the conditional variance, note E(�(jjl)) = 0 and recallthe notation from (4.9) �21(q) = Var(�(jjl) = 1c2Var(W q + (1�W )q):So the conditional variance of d(q; l) isE(d2(q; l)jFl�1) =E �Xjjl�1 l�1Yi=1W q(jji)c �(jjl)�2jFl�1!

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 13= Xjjl�1pjl�1 l�1Yi=1�W q(jji)c � l�1Yi=1�W q(pji)c �E(�(jjl)�(pjl)):Since �(pjl)) ? �(jjl)) if pjl 6= jjl we haveE(d2(q; l)jFl�1) = Xjjl�1�l�1Yi=1�W q(jji)c ��2�21(q)= Xjjl�1 l�1Yi=1�W 2q(jji)c(2q) �al�1(q)�21(q)=M(2q; l� 1)al�1(q)�21(q):Thus the conditional variance of M(q; l) islXi=1 E(d2(q; i)jFi�1) = lXi=1M(2q; i� 1)ai�1(q)�21(q):(6.3)Furthermore,E(d2(q; l)) = E(E(d2(q; l)jFl�1)) = EM(2q; l� 1)al�1(q)�21(q) = al�1(q)�21(q)and thus Var(M(q; l)) = lXi=1 E(d2(q; i)) = �21(q) lXi=1 ai�1(q):(6.4)This leads to the following facts.Proposition 6.2. If q < q� so that a(q) < 1, the martingale f(M(q; l);Fl); l � 0g is L2-boundedand hence uniformly integrable. It follows thatE(M(q;1) = 1; M(q; l) = E(M(q;1)jFl) ;(6.5)and M(q; l)! M(q;1) almost surely and in L2. Moreover, if q� � q < q�, then the martingalef(M(q; l);Fl); l � 0g is Lp-bounded for some 1 < p < 2 and, hence, still uniformly integrable,(6.5) still holds and M(q; l)!M(q;1) almost surely and in Lp.Remark. The proof will show that when q� � q < q�, we may take p = r0, where r0 is given inProposition 5.2 (iii). See (5.15).Proof. Suppose �rst that q < q�. We have from (6.4) thatsupl�0 E(M(q; l)� 1)2 =supl�0 Var(M(q; l))= liml!1 " lXi=1 E(d2(q; i)= 1Xi=1 ai�1(q)�21(q) <1:

14 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERThe rest follows from standard martingale theory (eg, [16, page 68], [18]).Now let q < q� and we consider uniform integrability without L2 boundedness. Suppose1 < p � 2 and for two paths j1 and j2 denote by mj1;j2 the largest i � l such that j1ji = j2ji.We haveE(M(q; l))p = 1c(q)lpE�Xj1jlXj2jl lYi=1W q(j1ji)W q(j2ji)�p=2� 1c(q)lp lXk=0E� Xj1jlj2jlmj1;j2=k lYi=1W q(j1ji)W q(j2ji)�p=2� 1clp(q) lXk=0E�Xjjk kYi=1W 2q(jji) �Xj(1)k+1;:::;j(1)lj(2)k+1;:::;j(2)lj(1)k+1 6=j(2)k+1 lYi=k+1W q(j1; : : : ; jk; j(1)k+1; : : : ; j(1)l ji)W q(j1; : : : ; jk; j(2)k+1; : : : ; j(2)l ji)�p=2� 1clp(q) lXk=0Xjjk E�W pq�k �E� Xj(1)k+1;:::;j(1)lj(2)k+1;:::;j(2)lj(1)k+1 6=j(2)k+1 lYi=k+1W q(j1; : : : ; jk; j(1)k+1; : : : ; j(1)l ji)W q(j1; : : : ; jk; j(2)k+1; : : : ; j(2)l ji)�p=2� lXk=0�c(pq)cp(q)�k 1c(l�k)p(q) ��E Xj(1)k+1;:::;j(1)lj(2)k+1;:::;j(2)lj(1)k+1 6=j(2)k+1 lYi=k+1W q(j1; : : : ; jk; j(1)k+1; : : : ; j(1)l ji)W q(j1; : : : ; jk; j(2)k+1; : : : ; j(2)l ji)�p=2= lXk=0 ap(q)k 1c(l�k)p(q)�E(W (1�W ))q(EW q)2(l�k�1)22(l�k�1) � 2�p=2=��E(W (1�W ))q�p=22p=2c(q)� lXk=0 ap(q)k:

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 15Here a product over the empty set is equal to 1. By Proposition 5.2 (iii) there is a p 2 (1; 2)such that ap(q) < 1. For this p the martingale f(M(q; l);Fl); l � 0g is Lp-bounded, and the restfollows, once again, from standard martingale theory.6.1. The distribution of M(q;1). The distribution of M(q;1) satis�es a simple recursionwhich can be used to derive additional information.Proposition 6.3. Suppose fM(q;1);M1(q;1);M2(q;1)g are iid with the same distribution asM(q;1), the martingale limit. Let W have the distribution of the cascade generator and supposeW and fM(q;1);M1(q;1);M2(q;1)g are independent. ThenM(q;1) d=W qM1(q;1)c(q) + (1�W )qM2(q;1)c(q)(6.6)and for any q > 0, P [M(q;1) = 0] = 0 or 1;(6.7)so that E(M(q;1)) = 1 implies P [M(q;1) = 0] = 0:Proof. We writeM(q;1) = liml!1Xjjl lYi=1W q(jji)cl= liml!1� Xj2;:::;jl lYi=1W q(0; j2; : : : ; jl)cl + Xj2;:::;jl lYi=1 W q(1; j2; : : : ; jl)cl �= liml!1�W q(0) Xj2;:::;jl lYi=2 W q(0; j2; : : : ; jl)cl + (1�W (0))q Xj2;:::;jl lYi=2W q(1; j2; : : : ; jl)cl �d=W q(0)M1(q;1)c + (1�W (0))qM2(q;1)c :Now we verify (6.7). De�ne p0 =P [M(q;1) = 0]pW (0) =P [W = 0] = P [W = 1]:Then since c(q) 6= 0p0 =P [M(q;1) = 0] = P [W qM1(q;1) + (1�W )qM2(q;1) = 0]=P [00;W = 0] + P [00;W = 1] + P [00; 0 < W < 1]:>From this we conclude p0 = 2pW (0)p0+ (1� 2pW (0))p20so that p0(1� 2pW (0)) = p20(1� 2pW (0)):If 0 < pW (0) < 1=2, then p0 = p20 and p0 = 0 or 1. If pW (0) = 1=2, then P [W = 0] = P [W =1] = 1=2 and W has a two point distribution and hence from Example 5 we know M(q; l) = 1which implies M(q;1) = 1.

16 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER7. Estimation: Subcritical ConsistencyWe propose two estimators of the structure function which depend on scaled summed powersof the wavelet coe�cients fZ(q; l); l � 1g. These are�̂1(q) = �̂1(q; l) = log2Z(q; l)l = log2P2l�1n=0 jd�l;njq � ql=2l(7.1) �̂2(q) = �̂2(q; l) = log2�Z(q; l + 1)Z(q; l) � = log2 P2l+1�1n=0 jd�(l+1);njq2q=2P2l�1n=0 jd�l;njq ! :(7.2)Analysis depends on showing that scaled versions of Z(q; l) are well-approximated by the mar-tingale and this is discussed next. Recall notational de�nitions (4.1), (4.2), (4.3), (4.6).Proposition 7.1. For q > 0, Z(q; l)clb �M(q; l) P! 0:If q 6= q�, the convergence is almost sure and if q < q�, the convergence is in L2. ThusZ(q; l)clb !M(q;1)(7.3)in the appropriate sense, depending on the case.Proof. Begin by writingZ(q; l)clb �M(q; l) =Xjjl lYi=1 W q(jji)c h j2W (jjl; 0)� 1jqb � 1i=Xjjl lYi=1 W q(jji)c �(jjl; 0)(7.4)where �(jjl; 0) ? �(pjl; 0) if jjl 6= pjl. Also E�(jjl; 0) = 0 and recall the notation from (4.10)�22(q) := E�2(jjl; 0) = 1b2Var(j2W � 1jq):If q < q�, so a(q) < 1, then similar to the calculations leading to (6.3) and (6.4) we �ndE�Z(q; l)clb �M(q; l)�2 =Xjjl Xpjl E lYi=1 W q(jji)c lYi=1W q(pji)c �(jjl; 0)�(pjl; 0)!=�22(q)(2EW 2q)lc2l(q) = �22(q)al(q)!0as l !1 since a(q) < 1. This shows the L2{convergence.For q > 0, the same method showsE��Z(q; l)clb �M(q; l)�2jFl� =�22(q)M(2q; l)al(q)

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 17=�22(q)Xjjl lYi=1 W 2q(jji)c2(q) =: �22(q)V (q; l);(7.5)and we need to show V (q; l) ! 0, almost surely as l ! 1. If the MKP condition fails, thenM(q; l)! 0 as l!1 and V (q; l) � �M(q; l)�2 ! 0:If the MKP condition holds, then from Proposition 5.2 (iii), there exists r0 2 (1; 2) such thatar0(q) < 1, and for p = r0=2 2 (1=2; 1) we have by the triangle inequality0 � V (q; l)p �Xjjl lYi=1W 2pq(jji)c2p(q) = (ar0(q))lM(r0q; l) a.s.! 0;(7.6)as l !1, since M(r0q;1) <1 almost surely.So in all cases V (q; l)! 0. For any � > 0; � > 0 we haveP [���Z(q; l)clb �M(q; l)��� > �jFl] =P [���Z(q; l)clb �M(q; l)��� > �jFl]1[V (q;l)�22(q)>�]+ P [���Z(q; l)clb �M(q; l)��� > �jFl]1[V (q;l)�22(q)��]�1[V (q;l)�22(q)>�] + ��2E �Z(q; l)clb �M(q; l)�2jFl!1[V (q;l)�22(q)��]=1[V (q;l)�22(q)>�] + ��2V (q; l)�22(q)1[V (q;l)�22(q)��]�1[V (q;l)�22(q)>�] + �2�2 :Take expectations and use V (q; l) a.s.! 0 and the arbitrariness of � to concludeZ(q; l)clb �M(q; l) P! 0;as l !1.For almost sure convergence, when q < q�, we get from (7.6) thatV (q; l) � �ar0(q)1=p�lM1=p(r0q; l)and so Pl V (q; l) <1 almost surely. Thus, for any � > 0,Xl P [jZ(q; l)clb �M(q; l)j > �jFl] � ��2XE �Z(q; l)clb �M(q; l)�2jFl! = (const)Xl V (q; l) <1;and by a generalization of the Borel-Cantelli lemma ([16, page 152]) we haveZ(q; l)clb �M(q; l) a.s.! 0:For q > q�, we prove almost sure convergence from Proposition 5.2 (iv) in a similar way.We use this comparison result Proposition 7.1 to get consistent estimators of the structurefunction �(q) in the subcritical case, by which we mean the case where the MKP condition holds.

18 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERProposition 7.2. De�ne �̂i(q) for i = 1; 2 by (7.1) and (7.2). Provided q < q�, so that the MKPcondition holds, both estimators are almost surely consistent for �(q):�̂i(q) a.s.! �(q); i = 1; 2;as l !1.Proof. In (7.3), take logarithms to the base 2 to getlog2 Z(q; l)� l log2 c(q)� log2 b! log2M(q;1);(7.7)almost surely as l!1. Divide through by l to get consistency of �̂1(q). To get the consistencyof �̂2(q), note from (7.7) thatlog2Z(q; l + 1)� log2 Z(q; l)� (l+ 1� l)�(q)! 0almost surely which proves consistency of �̂2(q).8. Subcritical Asymptotic Normality of Estimators.In this section we discuss second order properties of the estimators �̂i(q), i = 1; 2; de�nedin (7.1) and (7.2). The asymptotic normality for �̂1(q) requires a bias term which cannot beeliminated. This drawback, is eliminated by using �̂2(q), whose de�nition in terms of di�erencingremoves the bias term. However, take note of the suggestive remarks at the end of this Section8 about mean squared error.For this section it is convenient to write EFl and PFl for the conditional expectation andconditional probability with respect to the �-�eld Fl.8.1. Asymptotic Normality of �̂1(q). Begin by writingZ(q; l)clb �M(q; l) =Xjjl lYi=1 W q(jji)c h j2W (jjl; 0)� 1jqb � 1i(8.1) =:Xjjl Z(jjl)(8.2)where EFl(Z(jjl)) =0EFl(Z(jjl))2 =� lYi=1 W 2q(jji)c(2q) �al(q)�22(q);and recall �22(q) is de�ned in (4.10). Therefore,Xjjl EFl(Z(jjl))2 =M(2q; l)al(q)�22(q):(8.3)Our strategy for the central limit theorem is to regard Z(q;l)clb �M(q; l) as a sum of random variableswhich are conditionally independent given Fl and then apply the Liapunov condition ([18]) forasymptotic normality in a triangular array.

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 19Proposition 8.1. If 2q < q�, then as l !1PFl " Z(q;l)clb �M(q; l)pM(2q; l)al(q)�22(q) � x#! P [N(0; 1) � x] a.s.(8.4)where N(0; 1) is a standard normal random variable with mean 0 and variance 1. Taking expec-tations in (8.4) yields P " Z(q;l)clb �M(q; l)pM(2q; l)al(q)�22(q) � x#! P [N(0; 1) � x]:(8.5)Proof. By Proposition 5.2 (iii) there is � > 0 such that both 2q + � < q� anda1+�=2(2q) < 1:(8.6)Asymptotic normality in (8.4) will be shown if we establish the Liapunov conditionPjjl EFljZ(jjl)j2+�(M(2q; l)al(q))(2+�)=2 ! 0 a.s.;(8.7)where the denominator comes from (8.3). The numerator in the left side of (8.7) is boundedabove by EFlXjjl ����� lYi=1W q(jji)c �����2+� ��� j2W (jjl; 0)� 1jqb � 1���2+�=c1Xjjl � lYi=1 W q(2+�)(jji)c((2 + �)q) �cl((2 + �)q)cl(2+�)(q)=c1M((2 + �)q; l)(a2+�(q))l;where c1 = E��� j2W (jjl; 0)� 1jqb � 1���2+�:So the ratio in (8.7), apart from constants, is bounded byM((2 + �)q; l)(a2+�(q))lM(2q; l)1+�=2(a2(q))(1+�=2)l � M((2 + �)q;1)(a2+�(q))lM(2q;1)1+�=2(a2(q))(1+�=2)l :Note that the two random variables M((2+ �)q;1) and M(2q;1) are non zero with probability1 by Proposition 6.1. Check that a2+�(q)(a2(q))1+�=2 = a1+�=2(2q) < 1:So the Liapunov ratio is asymptotic to a �nite nonzero random variable times (a1+�=2(2q))l wherea1+�=2(2q) < 1 and the result is proven.Remark 8.1. In the denominator of (8.5) we may replace M(2q; l) by its limit M(2q;1). Thisfollows since almost surely 0 < M(2q;1) <1 for 2q < q� and thus Z(q;l)clb �M(q; l)pM(2q; l)al(q)�22(q) ;s M(2q; l)M(2q;1)!) (N(0; 1); 1)

20 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERby [2]. The desired result is obtained by multiplying components.Remark 8.2. Set Nl := Z(q;l)clb �M(q; l)pM(2q; l)al(q)�22(q) :(8.8)Then in R2, as l !1 (Nl; Nl+1)) �N1(0; 1); N2(0; 1)�;where Ni(0; 1); i = 1; 2 are iid standard normal random variables.To see this, write for any x; y 2 RP [Nl � x;Nl+1 � y] =EPFl+1 [Nl � x;Nl+1 � y]=E1[Nl�x]PFl+1 [Nl+1 � y]:By Proposition 8.1, PFl+1 [Nl+1 � y] = �(y) + �l(y) a.s.where �(y) is the standard normal cdf and where �l(y) L1! 0 and j�l(y)j � 2. SoP [Nl � x;Nl+1 � y] =E1[Nl�x]��(y) + �l(y)�=E1[Nl�x]�(y) + o(1)from the dominated convergence theorem, and hence we get convergence to!�(x)�(y):We now describe how this central limit behavior transfers to �̂1(q).Corollary 8.1. Under the assumptions in force in Proposition 8.1, we have��̂1(q)� �(q)�� l�1 log2 bM(q; l)pM(2q;1)al(q)�22(q)l log 2�M(q;l) ) N(0; 1):(8.9)Remark. The bias term l�1 log2�bM(q; l)� cannot be neglected.Proof. For brevity, write d(q) :=M(2q;1)al(q)�22(q);(8.10)and using the notation of (8.8) we haveZ(q; l) = clb�Nlpd(q) +M(q; l)�:Since �̂1(q) = 1l log2 Z(q; l);we have l�̂1(q) = l log2 c+ log2 b+ log2�Nlpd(q) +M(q; l)�and thus l��̂1(q)� �(q)� = log2 bM(q; l) + log2�1 + Nlpd(q)M(q; l) �:

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 21Since by (5.6) and assumption 2q < q� we have q < q�, we know that d(q) ! 0. Therefore,Nlpd(q)=M(q; l) ! 0, and the desired result follows by using the relation log(1 + x) � x forx # 0.The bias term in (8.9) is an unpleasant feature and thus we consider how to remove it bydi�erencing.8.2. Asymptotic Normality of �̂2(q). We now consider the asymptotic normality of �̂2(q). Itis possible to proceed from Proposition 8.1 but it turns out to be simpler to proceed with a directproof.Proposition 8.2. Suppose 2q < q�. Then�̂2(q)� �(q)pM(2q;1)al(q)�23(q)log 2�M(q;1) ) N(0; 1);(8.11)where �23(q) is de�ned in (4.11).Proof. Begin by observing thatZ(q; l + 1)cl+1b � Z(q; l)clb=Xjjl � lYi=1 W q(jji)c �"W q(jjl; 0)c j2W (jjl; 0; 0)� 1jqb + W q(jjl; 1)c j2W (jjl; 1; 0)� 1jqb� j2W (jjl; 0)� 1jqb #=:Xjjl � lYi=1 W q(jji)c �H(jjl)where we have setH(jjl) = W q(jjl; 0)c j2W (jjl; 0; 0)� 1jqb + W q(jjl; 1)c j2W (jjl; 1; 0)� 1jqb � j2W (jjl; 0)� 1jqb :(8.12)Check that EFlH(jjl) =12 + 12 � 1 = 0and EFlH2(jjl) =�23(q):It follows that conditionally on Fl we may treatZ(q; l + 1)cl+1b � Z(q; l)clb

22 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERas a sum of iid random variables with (conditional) varianceEFl�Z(q; l + 1)cl+1b � Z(q; l)clb �2 =Xjjl EFl�� lYi=1 W q(jji)c(q) �H(jjl)�2=Xjjl � lYi=1W 2q(jji)c(2q) �al(q)EH2(jjl)=M(2q; l)al(q)�23(q):As in the proof of Proposition 8.1 we may check the Liapunov condition and concludeZ(q;l+1)cl+1b � Z(q;l)clbpM(2q;1)al(q)�23(q) ) N(0; 1);or equivalently �c�1Z(q;l+1)Z(q;l) � 1�Z(q;l)clbpM(2q;1)al(q)�23(q) ) N(0; 1);and since Z(q; l)=(clb)!M(q;1) we have�c�1Z(q;l+1)Z(q;l) � 1�M(q;1)pM(2q;1)al(q)�23(q) ) N(0; 1):(8.13)Since c�1Z(q; l + 1)Z(q; l) � 1 P! 0;it follows that �̂2(q)� �(q) = log2�c�1Z(q; l + 1)Z(q; l) �=log�1 + �c�1Z(q;l+1)Z(q;l) � 1��log 2�c�1Z(q;l+1)Z(q;l) � 1log 2in probability. Combine this with (8.13) to complete the proof.For statistical purposes, the result (8.11) contains unobservables so as in [20, 17], considerationneeds to be given to replacing quantities which are not observed by observable estimators. Weassume that the random measure �1 is observed, or equivalently that the wavelet coe�cientsfd�l;ng are known. This means we have the quantities fZ(q; l)g.We de�ne the following useful observable quantityD2(q; l) =Xjjl lYi=1W 2q(jji)"W q(jjl; 0)j2W (jjl; 0; 0)� 1jqZ(q; l + 1) + W q(jjl; 1)j2W (jjl; 1; 0)� 1jqZ(q; l + 1)

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 23� j2W (jjl; 0)� 1jqZ(q; l) #2(8.14) =:Xjjl lYi=1W 2q(jji)V 2(jjl)=:Xjjl lYi=1W 2q(jji)" A+BZ(q; l + 1) � CZ(q; l)#2:Note that in this notation, H(jjl) = A+Bcb � Cb ;where H(jjl) is de�ned in (8.12). Recall also that EH2(jjl) = �23(q): In terms of the waveletcoe�cients we haveD2(q; l) =Xjjl " jd(�l; (jjl; 0))jq2�q(l+1)=2Z(q; l + 1) + jd(�l; (jjl; 1))jq2�q(l+1)=2Z(q; l + 1) � jd(�l; (jjl)jq2�ql=2Z(q; l) #2;(8.15)showing that D2(q; l) is an observable statistic.Corollary 8.2. Suppose 2q < q�. Then�̂2(q)� �(q)D(q; l)= log 2 ) N(0; 1)(8.16)as l !1.Proof. Because of (8.11), it su�ces to showD2(q; l)M(2q;1)al(q)�23(q)=M2(q;1) P! 1;as l !1. This is equivalent to showingZ2(q;l)c2l(q)b2(q)D2(q; l)M(2q; l)al(q)�23(q) P! 1:After some simple algebra, this ratio is the same asPjjlQli=1 W 2q(jji)c(2q)M(2q; l)�23(q) "A+Bbc Z(q; l)=clbZ(q; l + 1)=cl+1b!� Cb #2:Since M(2q; l) ! M(2q;1), it su�ces to show that the numerator converges in probability toM(2q;1)�23(q). Due to (7.3), we write the numerator asXjjl lYi=1W 2q(jji)c(2q) hA+Bbc � Cb i2 + op(1)2�A+ Bbc � Cb �+ op(1)2�A+ Bbc �2!=I + II + III:

24 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERQuantiles of standard normal

tau1

-2 -1 0 1 20.0

40.0

60.0

80.1

00.1

20.1

4 10121416

q = 0.75, shape = 1

Quantiles of standard normalta

u2

-2 -1 0 1 2-0

.05

0.0

0.0

50.1

0

10121416

q = 0.25, shape = 1

Figure 1. Normal QQ-plots of �̂1(q)� �(q) (left) and �̂2(q)� �(q) (right).As in Theorem 3.5 of [17],I !M(2q;1)E�A+ Bbc � Cb �2 =M(2q;1)�23(q);as desired. The terms I and II can readily be shown to go to 0.Figure 1 shows normal QQ plots of �̂i(q) � �(q), i = 1; 2, from simulated cascade data withbeta distributed cascade generator with shape parameter 1 (this makes the distribution uniform).The left plot is for q = 0:75 and the right is for q = 0:25. Each plot presents 4 graphs as thedepth l increases to 16. Note the better agreement of �̂2(q) to normality compared with �̂1(q).Concluding remarks on mean squared error: Examining Corollary 8.1 and Proposition 8.2yields that in the region 2q < q�, the conditional mean squared error of �̂1(q) is of the formO(1)p (al(q))2l2 + O(2)p (1)l2while that of �̂2(q) is O(3)p �al(q)�2.9. Supercritical Asymptotics; Lack of ConsistencyA critical issue with both the wavelet based estimator and the moment based ones used in [17],is that the asymptotic properties of the estimators are only valid in a certain range of q-values.For the wavelet estimators, we require q < q� for consistency and for the asymptotic normalityresults we require 2q < q�. We now show that the range q > q� is uninformative for our estimatorsand in fact our estimators are misleading when extended to inference for values beyond q�. Areliable estimate of q� would be valuable information. In place of such an estimate it is likelythat a graphical procedure is possible based on the following.

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 252 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

q = 8 Z/max.Z shape = 10

1012141618

Figure 2. Density plots of Z(q; l)=Z_(q; l).Let �̂_i (q) (i = 1; 2) have the same de�nition as �̂i(q) except that sum is replaced by max. Thuswe can de�ne by analogy with (4.3)Z_(q; l) =_jjl lYi=1W (jji)qj2W (jjl; 0)� 1jq:Note that Z_(q; l) = �Z_(1; l)�q:For large values of q, namely for q � q�, Z(q; l) is su�ciently well approximated by its largestsummand Z_(q; l): Figure 2 presents a density plot of simulated values of Z(q; l)=Z_(q; l) as thedepth l increases from 10 to 18; note the densities concentrate most mass around the point 1.Based on the idea of approximating Z(q; l) by Z_(q; l), since logZ_(q; l) = q logZ_(1; l) is linearin q, we anticipate that �̂1(q) should also be linear in q rendering �̂1(q) largely uninformative forinference purposes in the q � q�{region. A rough estimate of q� would be provided by the q-valuewhere the plots of �̂1(q) starts to look linear.Computer simulations o�er strong support for these remarks. Figure 3 shows overlaid simulatedvalues for �̂i(q); �̂_i (q); i = 1; 2 for large values of q. In the range of q{values beyond q� � 3:3, it isremarkable how linear the plots for �̂1(q) and �̂_1 (q) look and also how closely �̂_1 (q) approximates�̂1(q). Note the values in the plots have been multiplied by -1 to make the plots increasing.We now assume that q� <1 and examine this supercritical phenomenon when q � q� in moredetail. We will prove the asymptotic linearity of the estimator �̂1(q) for q � q�. In particular, theestimator �̂1(q) is not consistent when q > q�, and neither is the estimator �̂2(q).

26 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER

q

tau(

q)

0 2 4 6 8 10

-10

12

34 tau.1

tau.2theory

max.1max.2

levels = 16, q.step = 0.5, shape = 1

Figure 3. Plots of �̂i(q); �̂_i (q) for q� � 2:4.We start by introducing new notation. LetU(q; l) = c(q)lM(q; l) =Xjjl lYi=1W (jji)q; q > 0; l � 1 ;(9.1) U�(l) = maxjjl lYi=1W (jji); l � 1 ;(9.2)and de�ne, for q > 0, m(q) = lim supl!1 1l log2U(q; l) ;(9.3) m(q) = lim infl!1 1l log2U(q; l)as well as m� = lim supl!1 1l log2U�(l) ;(9.4) m� = lim infl!1 1l log2 U�(l) :

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 27It is immediate that for all q > 0 and 0 � � � q(U�(l))q � U(q; l) � (U�(l))� U(q � �; l) � 2l (U�(l))q :(9.5)In particular, for every q > 0 m(q)� 1 � qm� � m(q) ;(9.6) m(q)� 1 � qm� � m(q)almost surely.Note that it follows from Proposition 6.1 that for 0 < q < q�m(q) = m(q) = �(q) :(9.7)Since by the triangle inequality for all 0 < � < 1 and q > 0(U(q; l))� � U(�q; l) ;(9.8)we see that m(�q) � �m(q) :(9.9)For a q � q� and 0 < � < q�=q we hence getm(q) � 1�m(�q) = 1��(�q) ;and letting � " q�=q we conclude that for every q � q�m(q) � q �(q�)q� :(9.10)On the other hand, it follows from (9.5) that for all q > 0 and 0 � � � qm(q) � �m� +m(q � �) :Using (9.6) we obtain from here m(q1) � �m(q2)q2 +m(q1 � �)for all q1; q2 > 0 and 0 � � � q1. In particular, if 0 < q1 < q�, then for every 0 < q3 < q1 wechoose � = q1 � q3 and conclude, using (9.7), thatm(q2)q2 � m(q1)�m(q3)q1 � q3 = �(q1)� �(q3)q1 � q3 :Therefore, for all q > 0 m(q)q � sup0<p<q� � 0(p) :(9.11)However, sup0<p<q� � 0(p) = sup0<p<q� E(W q log2W )E(W q) � E(W q� log2W )E(W q�)= 1q� (1 + log2E(W q)) = �(q�)q�

28 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERby the de�nition of q�. Substituting into (9.11) immediately gives usm(q) � q �(q�)q�(9.12)for all q > 0. Comparing (9.12) with (9.10), we see thatm(q) := liml!1 1l log2U(q; l) = q �(q�)q�(9.13)for any q � q�. Moreover, using (9.6) with q !1 and (9.13) we immediately conclude thatm� := liml!1 1l log2U�(l) = �(q�)q� :(9.14)Remark 9.1. For the non{conservative cascades, for which the random variablesfW (jjl); j 2 f0; 1g1; l � 1gare iid, a statement analogous to (9.14) is equivalent to the so called �rst birth problem; see forinstance [9]. For the particular case of uniformly distributed W in the context of conservativecascades see also [11].We are now ready to establish the asymptotic behavior of the estimator �̂1(q) = �̂1(q; l) in thesupercritical case.Theorem 9.1. Let q � q�. Then, as l!1,�̂1(q; l)! q �(q�)q� a.s.(9.15)In particular, the estimator �̂1(q; l) is not a consistent estimator of �(q) if q > q�.Proof. Denote mZ(q) = lim supl!1 1l log2Z(q; l)and mZ(q) = lim infl!1 1l log2 Z(q; l) :Since Z(q; l) � U(q; l) for all q and l, we immediately conclude by (9.13) thatmZ(q) � m(q) = q �(q�)q� :(9.16)For the corresponding lower bound on mZ(q), note that since P (W 6= 1=2) > 0 and P (W = 0) <1=2 (otherwise q� =1), there is a � > 0 such thatp1 := P (j2W � 1j � �) > 0 and p2 := P (min(W; 1�W ) � �) > 0 :Let 0 < � < 1. Note that it follows from (9.14) that for all l large enough,P �U�(l) � 2(1��)l�(q�)=q�� � 12 :(9.17)For l � 1 let Nl = cardnjjl : W (jji) � � for all i = 1; : : : ; lo :

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 29By de�nition N0 = 1. Observe that for all l � 0Nl+1 = Nl +Ml ;where, given N0; N1; : : : ; Nl, the distribution of Ml is Binomial with parameters Nl and p2.Therefore, (Nl) is a supercritical branching process with progeny mean m = 1 + p2 > 1 andextinction probability 0. By Theorem I.10.3 of [1], page 30,liml!1 Nl(1 + p2)l = N̂ > 0 a.s. :(9.18)Let now 0 < � < 1. It follows by the de�nition of (Nl) that for every l � 1Z(l; q) � �[�l]q maxk=1;::: ;N[�l] U�k �l � [�l]�qj2W (l)k � 1jq ;(9.19)where �U�k �l � [�l]�; k � 1� are iid with the law of U��l � [�l]�and (W (l)k ; k � 1) are iid with the law of W :The two sequences are independent, and also independent of N[�l]. All the random variablesde�ned above can be assumed to be de�ned, for all l and k, on the same probability space(;F ; P ).We introduce several events. Let d = (1 + p2)1=2 > 1. Put1 = nNl � dl for all l large enougho :It follows from (9.18) that P (1) = 1. Let, further,(l)2 = d[�l][k=1nj2W (l)k � 1j � � and U�k �l � [�l]� � 2(1��)(l�[�l])�(q�)=q�o;l � 1. Note that by (9.17) we have P ((l)2 ) � 1�e�c�l for some c > 0 and all l � 1, and so letting2 = lim infl!1 (l)2 ;we see by Borel-Cantelli lemma that P (2) = 1. Therefore, P (1 \ 2) = 1 as well. However,for every ! 2 1 \ 2 we have by (9.19)Z(l; q) � �[�l]q2q(1��)(l�[�l])�(q�)=q��qfor all l large enough, which implies thatmZ(q) � q� log2 � + (1� �)(1� �)q �(q�)q� a.s. :Letting � ! 0 and �! 0 we conclude thatmZ(q) � q �(q�)q� :(9.20)Now the statement (9.15) follows from (9.16) and (9.20).

30 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGERFinally, it follows from part (iv) of Proposition 5.2 that�(q) > q�(q�)q�for all q > q�. Hence, the estimator �̂1(q; l) is not a consistent estimator of �(q) if q > q�.Here is an immediate corollary.Corollary 9.1. The estimator �̂2(q; l) is not a (strongly) consistent estimator of �(q) if q > q�.Proof. Notice that for every l � 1, �̂1(q; l) = 1l l�1Xj=0 �̂2(q; j) ;where Z(q; 0) = 1. Therefore, if for some q > q� �̂2(q; l) ! �(q) a.s. as l ! 1, then so does�̂1(q; l), which contradicts Theorem 9.1.An estimator related to �̂2(q; l) is�̂3(q; l) = log2�U(q; l+ 1)U(q; l) � := log2R(q; l); l � 1 :Since 1l log2U(q; l) = 1l l�1Xj=0 �̂3(q; j) ;(9.21)where U(q; 0) = 1, (9.13) and the same argument as that of Corollary 9.1 shows that �̂3(q; l)is not a strongly consistent estimator of �(q) if q > q� (even though it is a strongly consistentestimator of �(q) if q < q�). We can say more, however. Note that 0 � R(q; l) � 2 for all q andl. Furthermore, ER(q; l) = c(q) = 2�(q) for all q and l :Therefore, if for some q > q�, �̂3(q; l) converges a.s. to some limit �3(q) as l!1, then the 2�3(q)must have a �nite expectation equal to 2�(q). On the other hand, by (9.13) and (9.21) we musthave �3(q) equal to q�(q�)=q� a.s.. This contradiction shows that �̂3(q; l) cannot converge a.s. asl!1 if q > q�.We conjecture that the same is true for �̂2(q; l), in the sense that it does not converge a.s. asl ! 1 if q > q�. A possibility is that �̂2(q; l) converges in probability, and is weakly consistentfor q > q�. Whether or not this is true remains an open question.References[1] K. Athreya and P. Ney. Branching Processes. Springer-Verlag, New York, 1972.[2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.[3] A. Feldmann, A. C. Gilbert, and W. Willinger. Data networks as cascades: Investigating the multifractalnature of Internet WAN tra�c. In Proc. of the ACM/SIGCOMM'98, pages 25{38, Vancouver, B.C., 1998.[4] U. Frisch and G. Parisi. Fully developed turbulence and intermittancy. In M. Ghil, editor, Turbulence andPredictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland, Amsterdam, 1985.[5] A.C. Gilbert, W. Willinger, and A. Feldmann. Scaling analysis of conservative cascades, with applications tonetwork tra�c. IEEE Transactions on Information Theory, 45(3):971{991, 1999.

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 31[6] V. K. Gupta and E. C. Waymire. A statistical analysis of mesoscale rainfall as a random cascade. Journal ofApplied Meteorology, 32:251{267, 1993.[7] Richard Holley and Edward C. Waymire. Multifractal dimensions and scaling exponents for strongly boundedrandom cascades. Ann. Appl. Probab., 2(4):819{845, 1992.[8] J.P. Kahane and J. Peyri�ere. Sur certaines martingales de b. mandelbrot. Advances in Mathematics, 22:131{145, 1976.[9] J.F.C. Kingman. The �rst birth problem for an age{dependent branching process. The Annals of Probability,3:790{801, 1975.[10] A.N. Kolmogorov. Local structure of turbulence in an incompressible liquid for very large Reynolds numbers.C.R. Doklady Acad. Sci. URSS (N.S.), 30:299{303, 1941.[11] H. Mahmood. Evolution of Random Search Trees. Wiley, New York, 1992.[12] B. B. Mandelbrot. Intermittant turbulence in self-similar cascades: Divergence of high moments and dimensionof the carrier. Journal of Fluid Mechanics, 62:331{358, 1974.[13] B. B. Mandelbrot. Limit Lognormal Multifractal Measures. In Gotsman, Ne'eman, and Voronel, editors, TheLandau Memorial Conference, pages 309{340, Tel Aviv, 1990.[14] B.B. Mandelbrot. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer-Verlag, NewYork, 1998.[15] C. Meneveau and K. R. Sreenivasan. Simple multifractal cascade model for fully developed turbulence.Phys. Rev. Lett., 59:1424{1427, 1987.[16] J. Neveu. Discrete-Parameter Martingales, volume 10 of North-Holland Mathematical Library. North Holland,Amsterdam, 1975. Translated from the French original by T.P. Speed.[17] M. Ossiander and E. Waymire. Statistical estimation for multiplicative cascades. To appear; available fromfossiand,waymireg@math.orst.edu, 1999.[18] S.I. Resnick. A Probability Path. Birkh�auser, Boston, 1998.[19] R. H. Riedi and J. Levy-Vehel. Tcp tra�c is multifractal: A numerical study. Preprint, 1997.[20] Brent M. Troutman and Aldo V. Vecchia. Estimation of R�enyi exponents in random cascades. Bernoulli,5(2):191{207, 1999.School of Operations Research and Industrial Engineering and Department of Statistical Sci-ence, Cornell University, Ithaca, NY 14853E-mail address: sid@orie.cornell.edu, gennady@orie.cornell.eduAT&T Labs{Research, 180 Park Avenue, Florham Park, NJ 07932E-mail address: agilbert@research.att.com, walter@research.att.com